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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2507.21399 |
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| _version_ | 1866908824208670720 |
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| author | Hu, Yi |
| author_facet | Hu, Yi |
| contents | Laurent Lafforgue's presentation of a Grassmannian Gr$^{d, E}$ naturally comes equipped with the induced action of a subtorus $\mathbb{T}_\bullet$ of PGL$(E)$. By investigating the defining ideals of $\mathbb{T}_\bullet$-orbit closures through general points of Gr$^{d,E}$ and studying their degenerations, we obtain a morphsim $\mathfrak{q}: \mathbb{F}^{d, E_\bullet} \to \mathbb{H}^{d, E_{\bullet}}$ such that $\mathbb{H}^{d, E_\bullet}$, termed the $\mathfrak{G}$-quotient of Gr$^{d,E}$ by $\mathbb{T}_\bullet$, is birational to $[{\rm Gr}^{d, E}/\mathbb{T}_\bullet]$, and $\mathfrak{q}$, termed $\mathfrak{G}$-family of Gr$^{d,E}$ by $\mathbb{T}_\bullet$, is a family of general $\mathbb{T}_\bullet$-orbit closures and their degenerations. We obtain a series of new results on $\mathbb{H}^{d, E_{\bullet}}$ and $\mathbb{F}^{d, E_\bullet}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21399 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathfrak{G}$-Quotients of Grassmannians and Equations Hu, Yi Algebraic Geometry Laurent Lafforgue's presentation of a Grassmannian Gr$^{d, E}$ naturally comes equipped with the induced action of a subtorus $\mathbb{T}_\bullet$ of PGL$(E)$. By investigating the defining ideals of $\mathbb{T}_\bullet$-orbit closures through general points of Gr$^{d,E}$ and studying their degenerations, we obtain a morphsim $\mathfrak{q}: \mathbb{F}^{d, E_\bullet} \to \mathbb{H}^{d, E_{\bullet}}$ such that $\mathbb{H}^{d, E_\bullet}$, termed the $\mathfrak{G}$-quotient of Gr$^{d,E}$ by $\mathbb{T}_\bullet$, is birational to $[{\rm Gr}^{d, E}/\mathbb{T}_\bullet]$, and $\mathfrak{q}$, termed $\mathfrak{G}$-family of Gr$^{d,E}$ by $\mathbb{T}_\bullet$, is a family of general $\mathbb{T}_\bullet$-orbit closures and their degenerations. We obtain a series of new results on $\mathbb{H}^{d, E_{\bullet}}$ and $\mathbb{F}^{d, E_\bullet}$. |
| title | $\mathfrak{G}$-Quotients of Grassmannians and Equations |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2507.21399 |